# A Graph Isomorphism Condition and Equivalence of Reaction Systems

## TL;DR

This paper establishes a graph isomorphism condition for reaction systems, introducing the concept of skeletons to characterize when two systems exhibit equivalent global dynamics.

## Contribution

It introduces the notion of skeletons and provides necessary and sufficient conditions for their isomorphism, enabling the classification of reaction systems based on their dynamics.

## Key findings

- Skeletons uniquely define directed graphs of reaction systems.
- Necessary and sufficient conditions for graph isomorphism are established.
- Characterization of graphs representing reaction system dynamics is provided.

## Abstract

We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their corresponding transition graphs are isomorphic. We introduce the notion of a skeleton (a one-out graph) that uniquely defines a directed graph. We provide the necessary and sufficient conditions for two skeletons to define isomorphic graphs. This provides a necessary and sufficient condition for two reactions systems to be equivalent, as well as a characterization of the directed graphs that correspond to the global dynamics of reaction systems.